#33525

order	256
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 543 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,11,16)(7,17,21,45)(8,13,22,36)(9,25,29,43)(10,30,27,15)(12,32,35,19)(14,39,41,28)(18,44,46,66)(20,49,31,40)(23,52,50,65)(24,55,51,38)(26,42,57,34)(33,58,59,47)(37,62,61,54)(48,67,56,63)(53,64,68,60);(5,7,11,21)(6,12,16,35)(8,18,22,46)(9,26,29,57)(10,31,27,20)(13,33,36,59)(14,40,41,49)(15,42,30,34)(17,43,45,25)(19,39,32,28)(23,53,50,68)(24,56,51,48)(37,63,61,67)(38,64,55,60)(44,65,66,52)(47,62,58,54);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,33)(14,37)(15,38)(16,36)(17,44)(19,47)(20,48)(21,46)(25,52)(26,53)(27,51)(28,54)(29,50)(30,55)(31,56)(32,58)(34,60)(35,59)(39,62)(40,63)(41,61)(42,64)(43,65)(45,66)(49,67)(57,68);(8,22)(13,36)(18,46)(23,50)(24,51)(33,59)(37,61)(38,55)(44,66)(47,58)(48,56)(52,65)(53,68)(54,62)(60,64)(63,67)

Polycyclic presentation:

2^8|a^2=h,b^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,g^b=gh,d^c=dh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#138936, #138943, #138944, …
maximal normal subgroup of	#138936, #138943, #138944, …
maximal quotient of	#484202, #484548, #484627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26513);